Information Geometry and its Applications III

Abstract Shun-ichi Amari

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Shun-ichi Amari  (RIKEN, Japan)
Monday, August 02, 2010, room Hörsaal 1
Information geometry derived from divergence functions

Given a divergence function in a manifold, we can derive a Riemannian metric and a dual pair of affine connections, which are the essential constituents of Information Geometry. In the case of a family of probability distributions, its information-geometric structure is given from the invariance property. It consists of the Fisher information metric and the alpha connections. Moreover, the manifold of discrete probability distributions, that is the set of all probability distribution on a finite set, has a dually flat Riemannian structure. We study how these properties are related to the underlying divergence function. We define an invariant divergence in terms of information monotonicity, which leads us to the class of f-divergences. We then study a divergence function which gives dually flat affine structure. This is given by the Bregman divergence in terms of a convex function. The invariant and flat divergence in the manifold of probability distributions is the Kullback-Leibler divergence, and this is unique, but more generally it is the class of alpha-divergences in the manifold of positive measures. We can further discuss divergence functions in the manifold of positive-definite matrices, that of vision pictures and cones. A nonlinear transformation of a divergence function or a convex function causes a conformal change of the dual geometrical structure. In this context, we can discuss the dual geometry derived from the Tsallis or Renyi entropy. It again gives the dually flat structure to the family of discrete probability distributions or of positive measures. This can be extended to the family of positive-definite matrices. We can define the q-exponential family and related q-structure, which is a generalization of the current invariant information geometry.


Date and Location

August 02 - 06, 2010
University of Leipzig
04103 Leipzig

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
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Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
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František Matúš
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
Czech Republic
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Scientific Committee

Shun-ichi Amari
Brain Science Institute, Mathematical Neuroscience Laboratory
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
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Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
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Phone: (++49)-(0)341-9959-552
Fax: (++49)-(0)341-9959-555

05.04.2017, 12:42